The purpose of this study is to determine the long-term safety and effectiveness of an investigational drug, B2036-PEG, in patients with acromegaly, which is caused by growth hormone secreting tumors. This drug blocks the actions of growth hormone. This study will also determine if B2036-PEG is more effective than the two previously available drugs because they work by different mechanisms which are only effective in selected patients. All patients with acromegaly have high growth hormone levels and have growth hormone receptors. Therefore, this drug may have some effect in nearly all patients. To date, multiple studies in humans and animals have shown a decrease in one of the blood proteins (IGF-I) that is responsible for many of the signs and symptoms seen in acromegaly. Similar studies performed in humans at UNC and across the US have shown that the drug is well tolerated. This study will determine the long-term safety and effectiveness of different daily dosages during a long-term treatment regimen. The dose will be titrated in an attempt to normalized IGF-I values.